Trigonometria
Identidades Básicas
\(\tan x = \frac{\sin x}{\cos x}\)
\(\tan x = \frac{1}{\cot x}\)
\(\cot x = \frac{1}{\tan x}\)
\(\cot x = \frac{\cos x}{\sin x}\)
\(\sec x = \frac{1}{\cos x}\)
\(\csc x = \frac{1}{\sin x}\)
Identidades Pitagoreanas
\(\cos^2 x + \sin^2 x = 1\)
\(\sec^2 x - \tan^2 x = 1\)
\(\csc^2 x - \cot^2 x = 1\)
Identidades de Ângulo Duplo
\(\sin 2x = 2\sin x \cos x\)
\(\cos 2x = 1 - 2\sin^2 x\)
\(\cos 2x = 2\cos^2 x - 1\)
\(\cos 2x = \cos^2 x - \sin^2 x\)
\(\tan 2x = \frac{2\tan x}{1 - \tan^2 x}\)
Identidades de Soma/Diferença
\(\sin(s + t) = \sin s \cos t + \cos s \sin t\)
\(\sin(s - t) = \sin s \cos t - \cos s \sin t\)
\(\cos(s + t) = \cos s \cos t - \sin s \sin t\)
\(\cos(s - t) = \cos s \cos t + \sin s \sin t\)
\(\tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t}\)
\(\tan(s - t) = \frac{\tan s - \tan t}{1 + \tan s \tan t}\)
Identidades de Soma para Produto
\(\cos s \cos t = \frac{1}{2}[\cos(s - t) + \cos(s + t)]\)
\(\sin s \sin t = \frac{1}{2}[\cos(s - t) - \cos(s + t)]\)
\(\sin s \cos t = \frac{1}{2}[\sin(s + t) + \sin(s - t)]\)
\(\cos s \sin t = \frac{1}{2}[\sin(s + t) - \sin(s - t)]\)
Identidades de Ângulo Triplo
\(\sin 3x = -\sin^3 x + 3\cos^2 x \sin x\)
\(\sin 3x = -4\sin^3 x + 3\sin x\)
\(\cos 3x = \cos^3 x - 3\sin^2 x \cos x\)
\(\cos 3x = 4\cos^3 x - 3\cos x\)
\(\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}\)
\(\cot 3x = \frac{3\cot x - \cot^3 x}{1 - 3\cot^2 x}\)
Imagens de Função
\(y = \sin x \Rightarrow -1 \leq y \leq 1\)
\(y = \cos x \Rightarrow -1 \leq y \leq 1\)
\(y = \tan x \Rightarrow -\infty < y < \infty[/latex]
\(\)y = \cot x \Rightarrow -\infty < y < \infty[/latex]
\(\)y = \csc x \Rightarrow -\infty < y \leq -1 \cup 1 \leq y < \infty[/latex]
\(\)y = \sec x \Rightarrow -\infty < y \leq -1 \cup 1 \leq y < \infty[/latex]
\(\)y = \arcsin x \Rightarrow -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}\)
\(y = \arccos x \Rightarrow 0 \leq y \leq \pi\)
\(y = \arctan x \Rightarrow -\frac{\pi}{2} < y < \frac{\pi}{2}[/latex]
\(\)y = \text{arccot } x \Rightarrow 0 < x < \pi[/latex]
\(\)y = \text{arccsc } x \Rightarrow 0 \leq y < \frac{\pi}{2} \cup \pi \leq y < \frac{3\pi}{2}[/latex]
\(\)y = \text{arcsec } x \Rightarrow -\pi < y \leq -\frac{\pi}{2} \cup 0 < y < \frac{\pi}{2} < \infty[/latex]
Valores de Função
| x | sin(x) | cos(x) | tan(x) | cot(x) |
|---|---|---|---|---|
| \(\)0\) | \(0\) | \(1\) | \(0\) | Indefinido |
| \(\frac{\pi}{6}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{3}}{3}\) | \(\sqrt{3}\) |
| \(\frac{\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(1\) | \(1\) |
| \(\frac{\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\sqrt{3}\) | \(\frac{\sqrt{3}}{3}\) |
| \(\frac{\pi}{2}\) | \(1\) | \(0\) | Indefinido | \(0\) |
| \(\frac{2\pi}{3}\) | \(\frac{\sqrt{3}}{2}\) | \(-\frac{1}{2}\) | \(-\sqrt{3}\) | \(-\frac{\sqrt{3}}{3}\) |
| \(\frac{3\pi}{4}\) | \(\frac{\sqrt{2}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-1\) | \(-1\) |
| \(\frac{5\pi}{6}\) | \(\frac{1}{2}\) | \(-\frac{\sqrt{3}}{2}\) | \(-\frac{\sqrt{3}}{3}\) | \(-\sqrt{3}\) |
| \(\pi\) | \(0\) | \(-1\) | \(0\) | Indefinido |
| \(2\pi\) | \(0\) | \(1\) | \(0\) | Indefinido |